Question: Suppose $x-3$ and $y+3$ are multiples of $7$.

What is the smallest positive integer, $n,$ for which $x^2+xy+y^2+n$ is a multiple of $7$?
Solution: Since $x-3$ is a multiple of $7$, we know that $x\equiv 3\pmod 7$.

Since $y+3$ is a multiple of $7$, we know that $y\equiv -3\pmod 7$.

Therefore, \begin{align*}
x^2+xy+y^2+n &\equiv (3)^2 + (3)(-3) + (-3)^2 + n \\
&\equiv 9 - 9 + 9 + n \\
&\equiv 9 + n \qquad\pmod 7.
\end{align*}In other words, $9+n$ is a multiple of $7$. The smallest positive $n$ for which this is true is $n=\boxed{5}$.